Evaluate Definite Integrals


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Lecture Notes

The Fundamental Theorem of Calculus

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Section 3, Page 5 to page 6

Fundamental Theorem of Calculus is presented and justified.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Riemann Integrals (section 4 of lecture 14) and Derivatives (section 2 of lecture 1)
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Algorithm for Computing Riemann Integrals

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Section 4, Page 7

General algorithm for evaluating integrals using the Fundamental Theorem of Calculus.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Fundamental Theorem of Calculus (section 3 of this lecture)
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The Two Fundamental Theorems of Calculus (18.01, Fall 2006)

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Section 1, Page 1 to page 2

Statement and explanation of the First and Second Fundamental Theorems, with examples.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None
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Proof of the Two Fundamental Theorems (18.01, Fall 2006)

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Section 4, Page 6 to page 7

An intuitive argument justifying the Second Fundamental Theorem and a complete proof of the First Fundamental Theorem, including remarks about these theorems as they relate to definite and indefinite integrals.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: The Fundamental Theorems of Calculus (page 1 of this file)
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Online Textbook Chapter

The Fundamental Theorem

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Statement and explanation of the two forms for the Fundamental Theorem of Calculus.

Course: 18.013A Calculus with Applications, Spring 2005
Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Anti-derivatives (OT19.1) and Riemann Sums (OT20.2)
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Exam Questions

Functions Defined By Integrals (18.01, Fall 2006)

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Problem 4 (page 1)

Three part question finding bounds on the values of a function defined in terms of an integral and finding where the function is concave up or concave down.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF)# Page 1

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Fundamental Theorem of Calculus (18.01, Fall 2006)

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Problem 5 (page 1)

Using the Fundamental Theorem to evaluate an integral.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

 

Solution (PDF)# Page 1

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Functions Defined by Integrals (18.01, Fall 2006)

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Problem 3 (page 1)

Four part question which involves finding the derivative, critical points, and an estimation of a function which is defined in terms of an integral.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF)# Page 1

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Functions Defined by Integrals (18.01, Fall 2006)

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Problem 5 (page 2)

Four part question which involves finding critical points and values of a function which is defined in terms of an integral.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF)# Page 2

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Functions Defined by Integrals (18.01, Fall 2006)

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Problem 10 (page 2)

Finding the first and second derivatives of a function defined in terms of an integral, and expressing another integral in terms of that function.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF)# Page 1

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Fundamental Theorem of Calculus

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Problem 3C-1 (page 22) to problem 3C-6 (page 22)

Six questions which involve finding areas under curves and evaluating definite integrals.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF - 4.0 MB) Page 40

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Analysis of Graphs Limits of Functions Asymptotic & Unbounded Behavior Continuity: Property of Functions Parametric, Polar & Vector Function Concept of the Derivative Derivative at a Point Derivative as a Function Second Derivatives Applications of Derivatives Computation of Derivatives Interpretations & Properties of Definite Integrals Applications of Integrals Fundamental Theorem of Calculus Applications of Antidifferentiation Numerical Approximations to Definite Integ Concept of Series Series of Constants Taylor Series Graphs of Functions Intuitive Understanding: Limiting Process Calculating Limits Using Algebra Asymptotic Behavior: Infinity Continuity Continuity in Terms of Limits Analysis of Planar Curves Derivative Presented Graphically, Numerically & Analytically Derivative Interpreted as Instantaneous Rate of Change Derivative: Limit of the Difference Quotient Differentiability & Continuity Tangent Line - Curve at a Point & Local Linear Approximation Instantaneous Rate of Change Approximate Rate of Change Characteristics of f & f Relationship Between Behavior of f & Sign of f Mean Value Theorem & Geometric Consequences Characteristics: Graphs of f, f & f Relationship Between Concavity of f & Sign of f'' Points of Inflection - Concavity Changes Analysis of Curves Applications of Derivatives Optimization: Absolute & Relative Extrema Modeling Rates of Change Implicit Differentiation & Derivatives of Inverse Functions Derivative as a Rate of Change l'Hôpital's Rule Geometric Interpretation of Differential Equations Derivatives of Basic Functions Rules: Derivative of Sums, Products & Quotients of Functions Chain Rule & Implicit Differentiation Derivatives: Parametric, Polar & Vector Functions Definite Integral - Limit of Riemann Sums Rate of Change of a Quantity/Change of Quantity over Interval Properties of Definite Integrals Appropriate Integrals Evaluate Definite Integrals Representation of Specific Antiderivatives Antiderivatives From Derivatives of Basic Functions Integration by Substitution, Parts & Partial Fractions Improper Integrals Separable Equations & Modeling Riemann & Trapezoidal Sums Series, Convergence, Divergence Motivating Examples Geometric Series with Applications The Harmonic Series Alternating Series - Error Bound Series as Riemann Sums & the Integral Test Ratio Test - Convergence/Divergence Comparison Test Taylor Polynomial Approximation Maclaurin & Taylor Series Maclaurin Series for Basic Functions Manipulating Taylor Series Power Series Defining Functions Radius/Interval of Convergence Lagrange Error Bound